Engaging students: Introducing the number e

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Diana Calderon. Her topic, from Precalculus: introducing the number $e$.

How could you as a teacher create an activity or project that involves your topic?

– A project that I would want my students to work on that would introduce the number $e$ would be with having a weeklong project, assuming it is a block schedule, to allow the students to think about compound interest. the reason why we would use the compound interest formula to show $e$ is because, “It turns out that compounding weekly barely yields any more money than compounding monthly and at higher values of $n$, it gets closer and closer to what we recognize as the number $e$” The project would be about buying a car, the students would get to choose the car that they want, research multiple car dealerships, and they must figure out the calculations for compound interest in 24 months, 36 months, 48 months, 60 months and 72 months. For their final product they must have a picture/drawing of the car they chose to purchase, as well as choose the number of months they would like to finance for and the dealership they will purchase form. Finally, they must turn in a separate sheet with the calculations for the other months of finance they did not choose and why they chose not to choose them.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

– The number or better known as Euler’s number is a very important number in mathematics. Leonhard Euler was one of the greatest Swiss mathematicians from the 18th century. Although Euler was born in Switzerland, he spent much of his time in Russia and in Berlin. Euler’s father was great friends with Johan Bernoulli, who then became one of the most influential people in Euler’s life. Euler was also one who contributed to “ the mathematical notation in use today, such as the notation $f(x)$ to describe a function and the modern notation for the trigonometric functions”. Not only did Euler contribute to math, “He is also widely remembered for his contributions in mechanics, fluid dynamics, optics, astronomy, and music.” Euler was such an amazing mathematician that other mathematicians talked very highly of him such as Pierre-Simon Laplace who expressed how Euler is important in mathematics, “‘Read Euler, read Euler, he is master of us all’”

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?
– This video would be great to show to students to see how the number e is applied in different ways. The video starts off by talking about how we get $e$, “mathematically it is just what you get when you calculate 1 + (1/1000000)^1000000= 2.718 ≈ e and as the number gets bigger, you get Euler’s number, e=lim n→∞ f(n) (1+1/n)^n.” This is a really good video to show because the YouTuber talks about how when he was learning about the number e, he thought that it would never show up and then later realized that the better question was, when doesn’t it show up? He then proceeds to talk about how if you’re in high school then you start talking about it when it comes to compound interest, he then proceeds to give an example, “imagine you put $1 in a bank that pays out 100% interest per year, that means after one year you’ll have two dollars but that’s only if th interest compounds once a year. If instead it compounds twice a year you get 50% after 6 months and another 50% after 6 more months.”, and so on, he explains up to daily and compounding every second, nanosecond and so on, the amount in that persons bank would become$e ($2.1718). He then gives a real-world examples of probability with the number e. I would stop this video at 3:09 because that would give enough insight to the students about other applications of the number e and why it useful for them to learn it and not just think about it as a button in their calculator. Engaging students: Introducing the number e In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). This student submission comes from my former student Julie Thompson. Her topic, from Precalculus: introducing the number $e$. What interesting word problems using this topic can your students do now? I found a very interesting word problem involving the number e and derangements. A derangement is a permutation of a set in which no element is in its original place. The word problem I found is as follows: “At the bohemian jazz parties frequented by aficionados of the number e, the espresso flows freely, and at the end of the evening, party-goers are just as likely to go home in someone else’s overcoat as they are in their own. After such a party, what are the chances that at least one person goes home wearing the right coat?” To start off, we need to find out how many permutations, or how many combinations of ways the coats can be put on at random when guests leave the party. The problem asks us to identify the chance that at least one person IS wearing the right coat. In other words, we need to delete all the combinations in which nobody grabbed the correct coat. These are the derangements. Interestingly, when you divide the number of permutations by the number of derangements, you get a number extremely close to the value of e. And the ratio is always so. Looking at a numerical example with 10 guests, the number of ways 10 people can pick up 10 coats (permutations) is 3628800, and the number of ways nobody would pick up the right coat is 1334961. Dividing, 3628800/1334961= 2.71828, which is extremely close to e. Therefore, the chance of nobody getting the right coat is about 1 in e. How interesting. I feel like this word problem would really interest students! How was this topic adopted by the mathematical community? The number e was not discovered as ‘naturally’ as you may think. Mathematicians came close to discovering e in their calculations many times in the 17th century but thought it was just a random number without any real significance. The first person to calculate e is not documented, but historians believe it to not even be a mathematician, but a banker or trader. Why is this? The number e is very fundamental to a financial process that took off in the 17th century. “The number e lies at the foundations of one of the most fundamental processes of finance: compound interest.” Mathematicians, including Jacob Bernoulli, would later go on to define: . $e = \displaystyle \lim_{x \to \infty} \left(1 + \frac{1}{x} \right)^x$ “This is why the number e appears so often in modeling the exponential growth or decay of everything from bacteria to radioactivity.” This fact was adopted by the mathematical community and many mathematicians started collaborating and making many more discoveries on the number e, such as Euler, who estimated e correctly to 18 decimal places, gave the continued fraction expansion of e, and made a connection between e and the sine and cosine functions. The number e is one of the most beautiful and powerful number in all of mathematics and the use of it was adopted into mathematics most likely by a banker…how interesting. How can technology be used to effectively engage students with this topic? Any graphing technology, such as a TI calculator, Mathematica, MatLab, Desmos, etc., are great tools to use in order to engage students when discovering the number e. For instance, to convince students that the above limit is true, $e = \displaystyle \lim_{x \to \infty} \left(1 + \frac{1}{x} \right)^x$, I can have them graph the function for themselves and actually see that the function approaches the number e as x gets very large. Similarly, I can simulate numbers of e on a computer program with the expansion 1 + 1/1! + 1/2! + 1/3! + … to show the sum getting closer and closer to the value of e the more terms I add. I believe this will be really engaging because the expansion for the number e and the limit for e look like they have nothing to do with e at first glance. To make the connection between them graphically would be somewhat magical to students and hopefully make them curious for more. References: http://wmueller.com/precalculus/e/e6.html (this is word problem from A1) https://brilliant.org/wiki/the-discovery-of-the-number-e/ http://mathworld.wolfram.com/e.html http://www-history.mcs.st-and.ac.uk/HistTopics/e.html Powers Great and Small I enjoyed this reflective piece from Math with Bad Drawings about determining whether $a^b$ or $b^a$ is larger. The final answer, involving the number $e$, was a complete surprise to me. Short story: $e$ is the unique number so that $e^x > x^e$ for all positive $x$. Powers Great and Small Engaging students: Introducing the number e In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). This student submission comes from my former student Deanna Cravens. Her topic, from Precalculus: introducing the number $e$. How was this topic adopted by the mathematical community? The number e is a relatively newer irrational number if compared to pi. However, it first made its appearance very subtly in 1618. Napier was working on a table of natural logarithms, however it was not noted that the base was e. There were a few other appearances of e but mathematicians had not truly made a connection to it. Eventually in 1683, Jacob Bernoulli was looking at a business application dealing with continuously compounded interest and recognized that the log function and the exponential function were inverses. In 1690, a letter was written by Leibniz and e officially had a name, except it was called ‘b’ at the time. As it comes to no surprise, Euler had his hand in discovering e. He published Introductio in Analysin infinitorum in 1748 where he showed that e is the limit of $(1 + 1/n)^n$. Now Euler did not explicitly prove that e is irrational, however most people accepted it at that point, but it was indeed later proven. How could you as a teacher create an activity or project that involves your topic? Where does the number e come from? Well, the answer is a business application dealing with continuously compounded interest. However, students in a pre-calculus class can easily discover the number e without having to use the calculus behind it. Simply give students this short activity at the beginning of class. One of the good things about this activity is that it gives a brief snippet of the history of e before students begin to calculate it. Then, students can easily use a calculator and plug in the listed values in the table into the equation $(1+1/n)^n$. As the numbers get increasingly large, students will notice that they will all appear to be getting closer to 2.718… which is now known as the number e. As a teacher it is important to note that e is like pi, it is an irrational number that goes on forever and doesn’t have any sort of repeating pattern, yet it is extremely important in mathematics. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? This video would be excellent to show students who are asking, “why is e so important or where does it come from?” The video starts out by stating what e is approximately equal to. Then it gives a brief history about e and talks about compounded interest. It does a great job at explaining compounded interest. It is executed in a way where pre-calculus students can easily understand the concept. It also uses good visual cues to show how it would work. Next it lists several applications of e. These applications include: statistics through the normal curve, biology by modeling population growth, and physics by the exponential decay of a radioactive material. Overall, it does a great job showing the importance of e in real world applications. Thus, showing the importance of e to a pre-calculus students. Engaging students: Introducing the number e In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). This student submission comes from my former student Jillian Greene. Her topic, from Precalculus: introducing the number e. How does this topic extend what your students should have learned in previous courses? By this point in their mathematics career, the students have had plenty of experience with simple and compound interest formulas. Whether or not they discovered it them themselves through exploration in a class or their teacher just gave it to them, they’ve used it before. Now we can do an exploration activity that will connect that formula to the number e, and then to the limit. The activity will say: what if you invested$1 for 1 year at 100% compound interest? It’s a pretty good deal! But how much does the number of compounding periods affect the final value? Using the formula they have, A=P(1+r/n)^nt, they will calculate how much money they will make if it’s compounded:

• Yearly
• Biannually
• Quarterly
• Weekly
• Daily
• Hourly
• Every minute
• And every second

The first time it’s compounded, the final value will be $2. However, the more compounding periods you add, the closer to e you’ll get. For instance, weekly would be A=1(1+1/52)^52=2.69259695. Every second will get you A=1(1+1/31536000)^31536000=2.71828162, which is pretty to 2.718. The last three calculations will actually begin with 2.718. We can have some discussion with this as a class, bringing in the concept of limits. Then we can assess and see if anyone has seen this number before. If not, they can pop out their calculators and you can have them type “e” and then hit enter, and blow their minds. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? Though Euler does not receive credit for the first discovery of the number e, he does receive credit for naming it and first publishing it. Some say the e means exponential, some say he’d already published uses for a-d, and some say he named it after himself. He is quoted directly for saying “For the number whose logarithm is unity, let e be written, which is 2,7182817… [sic] whose logarithm according to Vlacq is 0,4342944… “ regarding the number e. He also has a couple of other choice quotes that illustrate his humor, ie “[upon losing the use of his right eye] ‘Now I will have less distraction.’” And “”Sir, hence God exists; reply!” In response to the French philosophe Diderot, who was trying to convert the court of Catherine the Great of Russia to atheism. Diderot had no idea what Euler was talking about and left the court to a chorus of laughter.” Back to e, however. If Euler did not first discover it, who did? A man name John Napier did the best he could to discover e. Napier was alive from 1550-1617, so he did not have access to a rich history of advanced algebra. Logarithm tables existed, some close to natural log, but none to identify this mystical number. Napier was merely trying to find an easier way to approach multiplication (and consequently exponentiation). His work, Construction of the Marvelous Rule of Logarithms, he states that X=Nap log y, where Nap log (107)=0. In today’s terms, with today’s math, we can translate that to Nap log y = 107 log1/e(y/107). How has this topic appeared in high culture (art, classical music, theatre, poetry* etc.)? After some discussion on this topic, if my class is a pre-AP or particularly curious class, I will have them go around and read this poem about e out loud. Then from this poem, I can have the students split up into groups. Each group will be responsible for dissecting this poem for certain things and then presenting their most interesting/exciting/relatable findings. One group will tackle the names; what history lesson is given to us here? Another group will handle applications; what did the various figures say we can do with e? The final group will report back on different representations of e; what all is e equal to? My expectations here would be for the students to see the insanely vast history and application of this number and gain some appreciation. I would expect to see Napier, Euler, and Leibniz for sure from the first group. From the second group, I would expect continuous compound interest, 1/e in probability and statistics, and calculus. The third group would be expected to present the numerical value of e, the limit that e is equal to, its infinite sum representation, and Euler’s identity. A number worthy of a 500 word poem and a slew of historical mathematicians must be important. The Enigmatic Number e by Sarah Glaz It ambushed Napier at Gartness, like a swashbuckling pirate leaping from the base. He felt its power, but never realized its nature. e‘s first appearance in disguise—a tabular array of values of ln, was logged in an appendix to Napier‘s posthumous publication. Oughtred, inventor of the circular slide rule, still ignorant of e‘s true role, performed the calculations. A hundred thirteen years the hit and run goes on. There and not there—elusive e, escape artist and trickster, weaves in and out of minds and computations: Saint-Vincent caught a glimpse of it under rectangular hyperbolas; Huygens mistook its rising trace for logarithmic curve; Nicolaus Mercator described its log as natural without accounting for its base; Jacob Bernoulli, compounding interest continuously, came close, yet failed to recognize its face; and Leibniz grasped it hiding in the maze of calculus, natural basis for comprehending change—but misidentified as b. The name was first recorded in a letter Euler sent Goldbach in November 1731: “e denontat hic numerum, cujus logarithmus hyperbolicus est=1.” Since a was taken, and Euler was partial to vowels, e rushed to make a claim—the next in line. We sometimes call e Euler‘s Number: he knew e in its infancy as 2.718281828459045235. On Wednesday, 6th of May, 2009, e revealed itself to Kondo and Pagliarulo, digit by digit, to 200,000,000,000 decimal places. It found a new digital game to play. In retrospect, following Euler‘s naming, e lifted its black mask and showed its limit: e=limn→∞(1+1n)ne=limn→∞(1+1n)n Bernoulli‘s compounded interest for an investment of one. Its reciprocal gave Bernoulli many trials, from gambling at the slot machines to deranged parties where nameless gentlemen check hats with butlers at the door, and when they leave, e‘s reciprocal hands each a stranger’s hat. In gratitude to Eulere showed a serious side, infinite sum representation: e=∑n=0∞1n!=10!+11!+12!+13!+⋯e=∑n=0∞1n!=10!+11!+12!+13!+⋯ For Euler‘s eyes alone, e fanned the peacock tail of e−12e−12’s continued fraction expansion, displaying patterns that confirmed its own irrationality. A century passed till e—through Hermite‘s pen, was proved to be a transcendental number. But to this day it teases us with speculations about ee. e‘s abstract beauty casts a glow on Euler’s Identity: eið + 1 = 0, the elegant, mysterious equation, where waltzing arm in arm with i and π, e flirts with complex numbers and roots of unity. We meet e nowadays in functional high places of CalculusDifferential EquationsProbabilityNumber Theory, and other ancient realms: y = ex e is the base of the unique exponential function whose derivative is equal to itself. The more things change the more they stay the same. e gathers gravitas as solid under integration, ∫exdx=ex+c∫exdx=ex+c a constant c is the mere difference; and often e makes guest appearances in Taylor series expansions. And now and then e stars in published poetry— honors and administrative duties multiply with age. References: http://www.maa.org/press/periodicals/convergence/the-enigmatic-number-iei-a-history-in-verse-and-its-uses-in-the-mathematics-classroom-the-annotated http://www.maa.org/publications/periodicals/convergence/napiers-e-napier http://www-history.mcs.st-and.ac.uk/HistTopics/e.html Computing e to Any Power: Index I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series examining one of Richard Feynman’s anecdotes about mentally computing $e^x$ for three different values of $x$. Part 1: Feynman’s anecdote. Part 2: Logarithm and antilogarithm tables from the 1940s. Part 3: A closer look at Feynman’s computation of $e^{3.3}$. Part 4: A closer look at Feynman’s computation of $e^{3}$. Part 5: A closer look at Feynman’s computation of $e^{1.4}$. Computing e to Any Power (Part 5) In this series, I’m exploring the following ancedote from the book Surely You’re Joking, Mr. Feynman!, which I read and re-read when I was young until I almost had the book memorized. One day at Princeton I was sitting in the lounge and overheard some mathematicians talking about the series for e^x, which is 1 + x + x^2/2! + x^3/3! Each term you get by multiplying the preceding term by x and dividing by the next number. For example, to get the next term after x^4/4! you multiply that term by x and divide by 5. It’s very simple. When I was a kid I was excited by series, and had played with this thing. I had computed e using that series, and had seen how quickly the new terms became very small. I mumbled something about how it was easy to calculate e to any power using that series (you just substitute the power for x). “Oh yeah?” they said. “Well, then what’s e to the 3.3?” said some joker—I think it was Tukey. I say, “That’s easy. It’s 27.11.” Tukey knows it isn’t so easy to compute all that in your head. “Hey! How’d you do that?” Another guy says, “You know Feynman, he’s just faking it. It’s not really right.” They go to get a table, and while they’re doing that, I put on a few more figures.: “27.1126,” I say. They find it in the table. “It’s right! But how’d you do it!” “I just summed the series.” “Nobody can sum the series that fast. You must just happen to know that one. How about e to the 3?” “Look,” I say. “It’s hard work! Only one a day!” “Hah! It’s a fake!” they say, happily. “All right,” I say, “It’s 20.085.” They look in the book as I put a few more figures on. They’re all excited now, because I got another one right. Here are these great mathematicians of the day, puzzled at how I can compute e to any power! One of them says, “He just can’t be substituting and summing—it’s too hard. There’s some trick. You couldn’t do just any old number like e to the 1.4.” I say, “It’s hard work, but for you, OK. It’s 4.05.” As they’re looking it up, I put on a few more digits and say, “And that’s the last one for the day!” and walk out. What happened was this: I happened to know three numbers—the logarithm of 10 to the base e (needed to convert numbers from base 10 to base e), which is 2.3026 (so I knew that e to the 2.3 is very close to 10), and because of radioactivity (mean-life and half-life), I knew the log of 2 to the base e, which is.69315 (so I also knew that e to the.7 is nearly equal to 2). I also knew e (to the 1), which is 2. 71828. The first number they gave me was e to the 3.3, which is e to the 2.3—ten—times e, or 27.18. While they were sweating about how I was doing it, I was correcting for the extra.0026—2.3026 is a little high. I knew I couldn’t do another one; that was sheer luck. But then the guy said e to the 3: that’s e to the 2.3 times e to the.7, or ten times two. So I knew it was 20. something, and while they were worrying how I did it, I adjusted for the .693. Now I was sure I couldn’t do another one, because the last one was again by sheer luck. But the guy said e to the 1.4, which is e to the.7 times itself. So all I had to do is fix up 4 a little bit! They never did figure out how I did it. My students invariably love this story; let’s take a look at the third calculation. Feynman knew that $e^{0.69315} \approx 2$, so that $e^{0.69315} e^{0.69315} = e^{1.3863} \approx 2 \times 2 = 4$. Therefore, again using the Taylor series expansion: $e^{1.4} = e^{1.3863} e^{0.0137} = 4 e^{0.0137}$ $\approx 4 \times (1 + 0.0137)$ $= 4 + 4 \times 0.0137$ $\approx 4.05$. Again, I have no idea how he put on a few more digits in his head (other than his sheer brilliance), as this would require knowing the value of $\ln 2$ to six or seven digits as well as computing the next term in the Taylor series expansion. Computing e to Any Power (Part 4) In this series, I’m exploring the following ancedote from the book Surely You’re Joking, Mr. Feynman!, which I read and re-read when I was young until I almost had the book memorized. One day at Princeton I was sitting in the lounge and overheard some mathematicians talking about the series for e^x, which is 1 + x + x^2/2! + x^3/3! Each term you get by multiplying the preceding term by x and dividing by the next number. For example, to get the next term after x^4/4! you multiply that term by x and divide by 5. It’s very simple. When I was a kid I was excited by series, and had played with this thing. I had computed e using that series, and had seen how quickly the new terms became very small. I mumbled something about how it was easy to calculate e to any power using that series (you just substitute the power for x). “Oh yeah?” they said. “Well, then what’s e to the 3.3?” said some joker—I think it was Tukey. I say, “That’s easy. It’s 27.11.” Tukey knows it isn’t so easy to compute all that in your head. “Hey! How’d you do that?” Another guy says, “You know Feynman, he’s just faking it. It’s not really right.” They go to get a table, and while they’re doing that, I put on a few more figures.: “27.1126,” I say. They find it in the table. “It’s right! But how’d you do it!” “I just summed the series.” “Nobody can sum the series that fast. You must just happen to know that one. How about e to the 3?” “Look,” I say. “It’s hard work! Only one a day!” “Hah! It’s a fake!” they say, happily. “All right,” I say, “It’s 20.085.” They look in the book as I put a few more figures on. They’re all excited now, because I got another one right. Here are these great mathematicians of the day, puzzled at how I can compute e to any power! One of them says, “He just can’t be substituting and summing—it’s too hard. There’s some trick. You couldn’t do just any old number like e to the 1.4.” I say, “It’s hard work, but for you, OK. It’s 4.05.” As they’re looking it up, I put on a few more digits and say, “And that’s the last one for the day!” and walk out. What happened was this: I happened to know three numbers—the logarithm of 10 to the base e (needed to convert numbers from base 10 to base e), which is 2.3026 (so I knew that e to the 2.3 is very close to 10), and because of radioactivity (mean-life and half-life), I knew the log of 2 to the base e, which is.69315 (so I also knew that e to the.7 is nearly equal to 2). I also knew e (to the 1), which is 2. 71828. The first number they gave me was e to the 3.3, which is e to the 2.3—ten—times e, or 27.18. While they were sweating about how I was doing it, I was correcting for the extra.0026—2.3026 is a little high. I knew I couldn’t do another one; that was sheer luck. But then the guy said e to the 3: that’s e to the 2.3 times e to the.7, or ten times two. So I knew it was 20. something, and while they were worrying how I did it, I adjusted for the .693. Now I was sure I couldn’t do another one, because the last one was again by sheer luck. But the guy said e to the 1.4, which is e to the.7 times itself. So all I had to do is fix up 4 a little bit! They never did figure out how I did it. My students invariably love this story; let’s take a look at the second calculation. Feynman knew that $e^{2.3026} \approx 10$ and $e^{0.69315} \approx 2$, so that $e^{2.3026} e^{0.69315} = e^{2.99575} \approx 10 \times 2 = 20$. Therefore, again using the Taylor series expansion: $e^3 = e^{2.99575} e^{0.00425} = 20 e^{0.00425}$ $\approx 20 \times (1 + 0.00425)$ $= 20 + 20 \times 0.00425$ $= 20.085$. Again, I have no idea how he put on a few more digits in his head (other than his sheer brilliance), as this would require knowing the values of $\ln 10$ and $\ln 2$ to six or seven digits as well as computing the next term in the Taylor series expansion: $e^3 = e^{\ln 20} e^{3 - \ln 20}$ $\approx 20 (1 +e^{ 0.0042677})$$\approx 20 \times \left(1 + 0.0042677 + \frac{0.0042677^2}{2!} \right)\$

$\approx 20.0855361\dots$

This compares favorably with the actual answer, $e^3 \approx 20.0855392\dots$.

Computing e to Any Power (Part 2)

In this series, I’m looking at a wonderful anecdote from Nobel Prize-winning physicist Richard P. Feynman from his book Surely You’re Joking, Mr. Feynman!. This story concerns a time that he computed $e^x$ mentally for a few values of $x$, much to the astonishment of his companions.

Part of this story directly ties to calculus.

One day at Princeton I was sitting in the lounge and overheard some mathematicians talking about the series for e^x, which is 1 + x + x^2/2! + x^3/3! Each term you get by multiplying the preceding term by x and dividing by the next number. For example, to get the next term after x^4/4! you multiply that term by x and divide by 5. It’s very simple.

When I was a kid I was excited by series, and had played with this thing. I had computed e using that series, and had seen how quickly the new terms became very small.

As noted, this refers to the Taylor series expansion of $e^x$, which is can be used to compute $e$ to any power. The terms get very small very quickly because of the factorials in the denominator, thus lending itself to the computation of $e^x$. Indeed, this series is used by modern calculators (with a few tricks to accelerate convergence). In other words, the series from calculus explains how the mysterious “black box” of a graphing calculator actually works.

Continuing the story…

“Oh yeah?” they said. “Well, then what’s e to the 3.3?” said some joker—I think it was Tukey.

I say, “That’s easy. It’s 27.11.”

Tukey knows it isn’t so easy to compute all that in your head. “Hey! How’d you do that?”

Another guy says, “You know Feynman, he’s just faking it. It’s not really right.”

They go to get a table, and while they’re doing that, I put on a few more figures.: “27.1126,” I say.

They find it in the table. “It’s right! But how’d you do it!”

For now, I’m going to ignore how Feynman did this computation in his head and instead discuss “the table.” The setting for this story was approximately 1940, long before the advent of handheld calculators. I’ll often ask my students, “The Brooklyn Bridge got built. So how did people compute $e^x$ before calculators were invented?” The answer is by Taylor series, which were used to produce tables of values of $e^x$. So, if someone wanted to find $e^{3.3}$, they just had a book on the shelf.

For example, the following page comes from the book Marks’ Mechanical Engineers’ Handbook, 6th edition, which was published in 1958 and which I happen to keep on my bookshelf at home.

Look down the fifth and sixth columns of this table, we see that $e^{3.3} \approx 27.11$. Somebody had computed all of these things (and plenty more) using the Taylor series, and they were compiled into a book and sold to mathematicians, scientists, and engineers.

But what if we needed an approximation better more accurate than four significant digits? Back in those days, there were only two options: do the Taylor series yourself, or buy a bigger book with more accurate tables.

Computing e to Any Power (Part 1)

Whenever I teach natural logarithms, I always share the following anecdote from the book Surely You’re Joking, Mr. Feynman!, which I read and re-read when I was young until I almost had the book memorized.

One day at Princeton I was sitting in the lounge and overheard some mathematicians talking about the series for e^x, which is 1 + x + x^2/2! + x^3/3! Each term you get by multiplying the preceding term by x and dividing by the next number. For example, to get the next term after x^4/4! you multiply that term by x and divide by 5. It’s very simple.

When I was a kid I was excited by series, and had played with this thing. I had computed e using that series, and had seen how quickly the new terms became very small.

I mumbled something about how it was easy to calculate e to any power using that series (you just substitute the power for x).

“Oh yeah?” they said. “Well, then what’s e to the 3.3?” said some joker—I think it was Tukey.

I say, “That’s easy. It’s 27.11.”

Tukey knows it isn’t so easy to compute all that in your head. “Hey! How’d you do that?”

Another guy says, “You know Feynman, he’s just faking it. It’s not really right.”

They go to get a table, and while they’re doing that, I put on a few more figures.: “27.1126,” I say.

They find it in the table. “It’s right! But how’d you do it!”

“I just summed the series.”

“Nobody can sum the series that fast. You must just happen to know that one. How about e to the 3?”

“Look,” I say. “It’s hard work! Only one a day!”

“Hah! It’s a fake!” they say, happily.

“All right,” I say, “It’s 20.085.”

They look in the book as I put a few more figures on. They’re all excited now, because I got another one right.

Here are these great mathematicians of the day, puzzled at how I can compute e to any power! One of them says, “He just can’t be substituting and summing—it’s too hard. There’s some trick. You couldn’t do just any old number like e to the 1.4.”

I say, “It’s hard work, but for you, OK. It’s 4.05.”

As they’re looking it up, I put on a few more digits and say, “And that’s the last one for the day!” and walk out.

What happened was this: I happened to know three numbers—the logarithm of 10 to the base e (needed to convert numbers from base 10 to base e), which is 2.3026 (so I knew that e to the 2.3 is very close to 10), and because of radioactivity (mean-life and half-life), I knew the log of 2 to the base e, which is.69315 (so I also knew that e to the.7 is nearly equal to 2). I also knew e (to the 1), which is 2. 71828.

The first number they gave me was e to the 3.3, which is e to the 2.3—ten—times e, or 27.18. While they were sweating about how I was doing it, I was correcting for the extra.0026—2.3026 is a little high.

I knew I couldn’t do another one; that was sheer luck. But then the guy said e to the 3: that’s e to the 2.3 times e to the.7, or ten times two. So I knew it was 20. something, and while they were worrying how I did it, I adjusted for the .693.

Now I was sure I couldn’t do another one, because the last one was again by sheer luck. But the guy said e to the 1.4, which is e to the.7 times itself. So all I had to do is fix up 4 a little bit!

They never did figure out how I did it.

My students invariably love this story.

In this series, I’d like to take a deeper look at this wonderful anecdote.